Comparing skew Schur functions: a quasisymmetric perspective Peter McNamara Bucknell University AMS/EMS/SPM International Meeting 11 June 2015 Slides and paper available from www.facstaff.bucknell.edu/pm040/ Comparing skew Schur functions quasisymmetrically Peter McNamara 1
Outline ◮ The background story: the equality question ◮ Conditions for Schur-positivity ◮ Quasisymmetric insights and the main conjecture Comparing skew Schur functions quasisymmetrically Peter McNamara 2
Preview F -support containment Dual of row overlap dominance Comparing skew Schur functions quasisymmetrically Peter McNamara 3
Schur functions Cauchy, 1815 ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) 4 1 3 3 ◮ Young diagram. 4 4 4 9 Example: 5 6 6 λ = ( 4 , 4 , 3 , 1 ) 7 Comparing skew Schur functions quasisymmetrically Peter McNamara 4
Schur functions Cauchy, 1815 ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) 1 3 3 4 ◮ Young diagram. 4 4 4 9 Example: 5 6 6 λ = ( 4 , 4 , 3 , 1 ) 7 ◮ Semistandard Young tableau (SSYT) Comparing skew Schur functions quasisymmetrically Peter McNamara 4
Schur functions Cauchy, 1815 ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) 1 3 3 4 ◮ Young diagram. 4 4 4 9 Example: 5 6 6 λ = ( 4 , 4 , 3 , 1 ) 7 ◮ Semistandard Young tableau (SSYT) The Schur function s λ in the variables x = ( x 1 , x 2 , . . . ) is then defined by � x # 1’s in T x # 2’s in T · · · . s λ = 1 2 SSYT T Example. = x 1 x 2 3 x 4 4 x 5 x 2 s 4431 6 x 7 x 9 + · · · . Comparing skew Schur functions quasisymmetrically Peter McNamara 4
Skew Schur functions Cauchy, 1815 ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) ◮ µ fits inside λ . 4 ◮ Young diagram. 4 4 9 Example: 5 6 6 λ/µ = ( 4 , 4 , 3 , 1 ) / ( 3 , 1 ) 7 ◮ Semistandard Young tableau (SSYT) The skew Schur function s λ/µ in the variables x = ( x 1 , x 2 , . . . ) is then defined by � x # 1’s in T x # 2’s in T · · · . s λ/µ = 1 2 SSYT T Example. x 3 4 x 5 x 2 s 4431 / 31 = 6 x 7 x 9 + · · · . Comparing skew Schur functions quasisymmetrically Peter McNamara 4
The beginning of the story s A : the skew Schur function for the skew shape A . Key Facts. ◮ s A is symmetric in the variables x 1 , x 2 , . . . . ◮ The (non-skew) s λ form a basis for the symmetric functions. Comparing skew Schur functions quasisymmetrically Peter McNamara 5
The beginning of the story s A : the skew Schur function for the skew shape A . Key Facts. ◮ s A is symmetric in the variables x 1 , x 2 , . . . . ◮ The (non-skew) s λ form a basis for the symmetric functions. Wide Open Question. When is s A = s B ? Determine necessary and sufficient conditions on shapes of A and B . = = Comparing skew Schur functions quasisymmetrically Peter McNamara 5
The beginning of the story s A : the skew Schur function for the skew shape A . Key Facts. ◮ s A is symmetric in the variables x 1 , x 2 , . . . . ◮ The (non-skew) s λ form a basis for the symmetric functions. Wide Open Question. When is s A = s B ? Determine necessary and sufficient conditions on shapes of A and B . = = ◮ Lou Billera, Hugh Thomas, Steph van Willigenburg (2004) ◮ John Stembridge (2004) ◮ Vic Reiner, Kristin Shaw, Steph van Willigenburg (2006) ◮ McN., Steph van Willigenburg (2006) ◮ Christian Gutschwager (2008) Comparing skew Schur functions quasisymmetrically Peter McNamara 5
Necessary conditions for equality Comparing skew Schur functions quasisymmetrically Peter McNamara 6
Necessary conditions for equality General idea: the overlaps among rows must match up. Comparing skew Schur functions quasisymmetrically Peter McNamara 6
Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. For a skew shape A , let overlap k ( i ) be the number of columns occupied in common by rows i , i + 1 , . . . , i + k − 1. Then rows k ( A ) is the weakly decreasing rearrangement of ( overlap k ( 1 ) , overlap k ( 2 ) , . . . ) . Example. A = Comparing skew Schur functions quasisymmetrically Peter McNamara 6
Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. For a skew shape A , let overlap k ( i ) be the number of columns occupied in common by rows i , i + 1 , . . . , i + k − 1. Then rows k ( A ) is the weakly decreasing rearrangement of ( overlap k ( 1 ) , overlap k ( 2 ) , . . . ) . Example. A = ◮ overlap 1 ( i ) = length of the i th row. Thus rows 1 ( A ) = 44211. Comparing skew Schur functions quasisymmetrically Peter McNamara 6
Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. For a skew shape A , let overlap k ( i ) be the number of columns occupied in common by rows i , i + 1 , . . . , i + k − 1. Then rows k ( A ) is the weakly decreasing rearrangement of ( overlap k ( 1 ) , overlap k ( 2 ) , . . . ) . Example. A = ◮ overlap 1 ( i ) = length of the i th row. Thus rows 1 ( A ) = 44211. ◮ overlap 2 ( 1 ) = 2, overlap 2 ( 2 ) = 3, overlap 2 ( 3 ) = 1, overlap 2 ( 4 ) = 1, so rows 2 ( A ) = 3211. Comparing skew Schur functions quasisymmetrically Peter McNamara 6
Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. For a skew shape A , let overlap k ( i ) be the number of columns occupied in common by rows i , i + 1 , . . . , i + k − 1. Then rows k ( A ) is the weakly decreasing rearrangement of ( overlap k ( 1 ) , overlap k ( 2 ) , . . . ) . Example. A = ◮ overlap 1 ( i ) = length of the i th row. Thus rows 1 ( A ) = 44211. ◮ overlap 2 ( 1 ) = 2, overlap 2 ( 2 ) = 3, overlap 2 ( 3 ) = 1, overlap 2 ( 4 ) = 1, so rows 2 ( A ) = 3211. ◮ rows 3 ( A ) = 11. Comparing skew Schur functions quasisymmetrically Peter McNamara 6
Necessary conditions for equality General idea: the overlaps among rows must match up. Definition [Reiner, Shaw, van Willigenburg]. For a skew shape A , let overlap k ( i ) be the number of columns occupied in common by rows i , i + 1 , . . . , i + k − 1. Then rows k ( A ) is the weakly decreasing rearrangement of ( overlap k ( 1 ) , overlap k ( 2 ) , . . . ) . Example. A = ◮ overlap 1 ( i ) = length of the i th row. Thus rows 1 ( A ) = 44211. ◮ overlap 2 ( 1 ) = 2, overlap 2 ( 2 ) = 3, overlap 2 ( 3 ) = 1, overlap 2 ( 4 ) = 1, so rows 2 ( A ) = 3211. ◮ rows 3 ( A ) = 11. ◮ rows k ( A ) = ∅ for k > 3. Comparing skew Schur functions quasisymmetrically Peter McNamara 6
Necessary conditions for equality Theorem [RSvW, 2006]. Let A and B be skew shapes. If s A = s B , then rows k ( A ) = rows k ( B ) for all k . Comparing skew Schur functions quasisymmetrically Peter McNamara 7
Necessary conditions for equality Theorem [RSvW, 2006]. Let A and B be skew shapes. If s A = s B , then rows k ( A ) = rows k ( B ) for all k . supp s ( A ) : Schur support of A supp s ( A ) = { λ : s λ appears in Schur expansion of s A } Example. A = s A = s 3 + 2 s 21 + s 111 supp s ( A ) = { 3 , 21 , 111 } . Comparing skew Schur functions quasisymmetrically Peter McNamara 7
Necessary conditions for equality Theorem [RSvW, 2006]. Let A and B be skew shapes. If s A = s B , then rows k ( A ) = rows k ( B ) for all k . supp s ( A ) : Schur support of A supp s ( A ) = { λ : s λ appears in Schur expansion of s A } Example. A = s A = s 3 + 2 s 21 + s 111 supp s ( A ) = { 3 , 21 , 111 } . Theorem [McN., 2008]. It suffices to assume supp s ( A ) = supp s ( B ) . Comparing skew Schur functions quasisymmetrically Peter McNamara 7
Necessary conditions for equality Theorem [RSvW, 2006]. Let A and B be skew shapes. If s A = s B , then rows k ( A ) = rows k ( B ) for all k . supp s ( A ) : Schur support of A supp s ( A ) = { λ : s λ appears in Schur expansion of s A } Example. A = s A = s 3 + 2 s 21 + s 111 supp s ( A ) = { 3 , 21 , 111 } . Theorem [McN., 2008]. It suffices to assume supp s ( A ) = supp s ( B ) . Converse is definitely not true: � = Comparing skew Schur functions quasisymmetrically Peter McNamara 7
Schur-positivity order Our interest: inequalities. Skew Schur functions are Schur-positive: � c λ s λ/µ = µν s ν . ν Original Question. When is s λ/µ − s σ/τ Schur-positive? Comparing skew Schur functions quasisymmetrically Peter McNamara 8
Schur-positivity order Our interest: inequalities. Skew Schur functions are Schur-positive: � c λ s λ/µ = µν s ν . ν Original Question. When is s λ/µ − s σ/τ Schur-positive? Definition. Let A , B be skew shapes. We say that A ≥ s B s A − s B if is Schur-positive. Original goal: Characterize the Schur-positivity order ≥ s in terms of skew shapes. Comparing skew Schur functions quasisymmetrically Peter McNamara 8
Example of a Schur-positivity poset If B ≤ s A then | A | = | B | . Call the resulting ordered set P n . Then P 4 : Comparing skew Schur functions quasisymmetrically Peter McNamara 9
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